We define non-dimensionalised density \(\widetilde{\rho }\) , radial velocity \(\widetilde{u}\) , temperature \(\widetilde{T}\) , and pressure \(\widetilde{P}\)  fields. From dimensional analysis [1]},
\(\rho (\vec{r},t)&= \rho _0 \widetilde{\rho }(\xi ),\\u(\vec{r},t)&=\frac{r}{t} \widetilde{u}(\xi ),\\T(\vec{r},t)&=\frac{r^2}{t^2}\widetilde{T}(\xi ),\\p(\vec{r},t)&=\frac{\rho _0 r^2}{t^2} \widetilde{P}(\xi ),\) 
